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### The polynomial system of equations

The coefficients of polynomials from the cascade in [4] are polynomials in terms of the cosines and sines of the angles. The flatness equations for the polynomials are linear combinations of the coefficients of the polynomials .This shows the principle leading to the use of techniques for solving polynomial systems in this signal processing context.

Actually the straightforward application of existing algorithms to the polynomial system obtained by writing the flatness equations in terms of the cascade parameters cannot design filters with support larger than 66 and 2 degrees of flatness. So to design filter banks with higher regularity we propose a change of variables, splitting the system into two smaller subsystems, which make it possible to design filters with support 1616 and 5 degrees of flatness.

The cascade form we use in the sequel, borrowed from [4], and the formulation of the issue, maximizing the flatness of filters of given size, turns it into a polynomial system.